Relationship between R.E., co-R.E. and Recursive Languages:

Theorem:  Let L be any language. L is R.E. and co-R.E. iff L is recursive. Proof:

• (→) Let L be R.E. and co-R.E. Then, 'L is R.E. Thus, L is recursive.
• (←) Let L be recursive. Then, L is R.E. From closure under complementation of the class of recursive languages, L is recursive. Then, 'L is also R.E.
• Therefore, L is co-R.E.

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